Asymptotic Investigation of Star-Shaped Message Switching Networks with a Large Number of Radial Rays

The paper considers networks consisting of a large number of independent message sources and a central point that is connected to each source by two-way communication lines. When a message from a source arrives at the central point, it is instantaneously switched to the input of the communications line that leads from the central point to another source, inconformity with the address of the message. It is assumed that each source generates a homogeneous Poisson flow of messages whose lengths are mutually independent, exponentially distributed, and independent of the ensemble of instants of appearance of messages at the sources. The message length remains unaltered as the message passes through the system. The authors investigate the asymptotic behavior of the queues at the central point when the addresses are uniformly distributed and the number of sources increases without limit. It is shown that the waiting time of each message in queue at the input of the line leading from the source to the central point, and its waiting time at the central point, are independent identically distributed random quantities in the limit. From this, asymptotic formulas are derived for the probability characteristics of the communications network in question.